Date: Nov 30, 2012 1:17 PM
Author: Zaljohar@gmail.com
Subject: What are sets? again
The following is an account about what sets are, first I'll write theexposition of this base theory in brief, then I'll discuss somerelated issues.

Language: FOL + P, Rp

P stands for "is part of"

Rp stands for "represents"

Axioms: Identity theory axioms +

I. Part-hood: P partially orders the universe.

ll. Supplementation: x P y & ~ y P x -> Exist z. z P y & ~ x P z.

Def.) atom(x) <-> for all y. y P x -> x P y

Def.) x atom of y <-> atom(x) & x P y.

Def.) c is a collection of atoms iff for all y. y P c -> Exist z. zatom of y.

Def.) c is atomless <-> ~ Exist x. x atom of c

lll. Representation: x Rp c & y Rp d -> (x=y<->c=d)

lV. Representatives: x Rp c -> atom(x)

V. Null: Exist! x. (Exist c. x Rp c & c is atomless).

A Set is an atom that uniquely represents a collection of atoms orabsence of atoms.

This theory can interpret second order arithmetic. And I like to thinkof it as a base theory on top of which any stronger set theory canhave its axioms added to it relativized to sets and with setmembership defined as above, so for example one can add all ZFC axiomsin this manner, and the result would be a theory that defines a modelof ZFC, and thus proves the consistency of ZFC. Anyhow this would onlybe a representation of those theories in terms of differentprimitives, and it is justified if one think of those primitives as amore natural than membership, or if one think that it is useful toexplicate the later. Moreover this method makes one see the WholeOntology involved with set\class theories, thus the bigger picturerevealed! This is not usually seen with set theories or even classtheories as usually presented, here one can see the interplay betweensets and classes (collections of atoms), and also one can easily addUr-elements to this theory and still be able to discriminate it fromthe empty set at the same time, a simple approach is to stipulate theexistence of atoms that do not represent any object. It is also veryeasy to explicate non well founded scenarios here in almost flawlessmanner. Even gross violation of Extensionality can be easilycontemplated here. So most of different contexts involved with variousmaneuvering with set\class theories can be easilyparalleled here and understood in almost naive manner.

In simple words the above approach speaks about sets as being atomicrepresentatives of collections (or absence) of atoms, the advantage isclearly of obtaining a hierarchy of objects. Of course an atom hererefers to indivisible objects with respect to relation P here, andthis is just a descriptive atom-hood that depends on discourse of thistheory, it doesn't mean true atoms that physically have no parts, itonly means that in the discourse of this theory thereis no description of proper parts of them, so for example one can addnew primitive to this theory like for example the primitive "physical"and stipulate that any physical object is an atom, so a city forexample would be an atom, it means it is descriptively an atom as faras the discourse of this theory is concerned, so atom-hood is adescriptive modality here. From this one can understand that a set isa way to look at a collection of atoms from atomic perspective, so theset is the atomic representative of that collection, i.e. it is whatone perceives when handling a collection of atoms as one descriptive\discursive whole, this one descriptive\discursive whole is actuallythe atom that uniquely represents that collection of atoms, and thecurrent methodology is meant to capture this concept.

Now from all of that it is clear that Set and Set membership are notpure mathematical concepts, they are actually reflecting ahierarchical interplay of the singular and the plural, which is at amore basic level than mathematics, it is down at the level of Logicactually, so it can be viewed as a powerful form of logic, even theadded axioms to the base theory above like those of ZFC are reallymore general than being mathematical and even when mathematicalconcepts are interpreted in it still the interpretation is notcompletely faithful to those concepts. However this powerful logicalbackground does provide the necessary Ontology required formathematical objects to be secured and fortheir rules to be checked for consistency.

But what constitutes mathematics? Which concepts if interpreted in theabove powerful kind of logic would be considered as mathematical? Thisproves to be a very difficult question. I'm tending to think thatmathematics is nothing but "Discourse about abstract structure", whereabstract structure is a kind of free standing structural universal.Anyhow I'm not sure of the later. I don't think anybody reallysucceeded with carrying along such concepts.